Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling

Abstract

This article presents a statistical theory for texture modeling. This theory combines filtering theory and Markov random field modeling through the maximum entropy principle, and interprets and clarifies many previous concepts and methods for texture analysis and synthesis from a unified point of view. Our theory characterizes the ensemble of images \(I\) with the same texture appearance by a probability distribution \(f(I)\) on a random field, and the objective of texture modeling is to make inference about \(f(I)\), given a set of observed texture examples. In our theory, texture modeling consists of two steps. (1) A set of filters is selected from a general filter bank to capture features of the texture, these filters are applied to observed texture images, and the histograms of the filtered images are extracted. These histograms are estimates of the marginal distributions of \(f(I)\). This step is called feature extraction. (2) The maximum entropy principle is employed to derive a distribution \(p(I)\), which is restricted to have the same marginal distributions as those in (1). This \(p(I)\) is considered as an estimate of \(f(I)\) . This step is called feature fusion. A stepwise algorithm is proposed to choose filters from a general filter bank. The resulting model, called FRAME (Filters, Random fields And Maximum Entropy), is a Markov random field (MRF) model, but with a much enriched vocabulary and hence much stronger descriptive ability than the previous MRF models used for texture modeling. Gibbs sampler is adopted to synthesize texture images by drawing typical samples from \(p(I)\) , thus the model is verified by seeing whether the synthesized texture images have similar visual appearances to the texture images being modeled. Experiments on a variety of 1D and 2D textures are described to illustrate our theory and to show the performance of our algorithms. These experiments demonstrate that many textures which are previously considered as from different categories can be modeled and synthesized in a common framework.